TITLE: Topological "Freshman's Dream"
QUESTION [28 upvotes]: When one learns about quotient and product spaces in topology for the first time, it is perhaps natural to expect that they would behave like mutual inverses:

Topological Freshman's Dream (TFD). For a space $X$ and subspace $\emptyset \neq Y\subseteq X$, the spaces $(X/Y)\times Y$ and $X$ are homeomorphic.

It is not too hard to see that TFD is not true even for very simple spaces. For example, pick $X=[0,1]$ and $Y=\{0,1\}$, then $X/Y\times Y$ is the disjoint union of two copies of $S^1$, obviously not the same as $X$.
There are two trivial cases when TFD does hold, when $Y$ is a single point and when $Y$ is the whole space $X$.

Q. Is there any nontrivial example when TFD holds?

I've tried for a while to construct such an example without success.

Some incomplete observations:

If $X$ is connected, then $Y$ must be as well. Otherwise, $(X/Y)\times Y$ would be disconnected.
We can apply the tools of algebraic topology to see, for example, that TFD implies $\pi_n(X)\cong\pi_n(X/Y)\times\pi_n(Y)$. This condition is quite hard to satisfy since it implies that the homotopy groups of the quotient space $X/Y$ are simpler than that of $X$, which generally fails quite spectatularly. A similar idea can also be applied to the homology and cohomology groups.
A special case of the above point implies that if $X$ is simply connected, then both $Y$ and $X/Y$ are simply connected (take the fundamental group $\pi_1$).

Any input is appreciated!

REPLY [4 votes]: Here is a connected compact Hausdorff space with this property: the Hilbert cube $X:=[0,1]^\omega$. We can choose $Y:=\{x\in [0,1]^\omega\mid x_1=0\}\subset X$.
Admittably, this is still an "Eric Wofsey type example": it satisfies $X/Y\cong X$ and $X\times Y\cong X$, and now I wonder whether there are example that satisfy neither.

Proving $X/Y\cong X$
The Hilbert cube can be identified with the following convex compact subset of $ \ell^2$:
$$X:=[0,1]\times[0,1/2]\times[0,1/3]\times\cdots.$$
I believe that the quotient $X/Y$ is homeomorphic to the following subset of $X$:
$$X/Y\cong \Big\{x\in X \;\Big\vert\; x_i\le \frac{x_1}i\text{ for all $i\ge 2$}\Big\}.$$
I furthermore believe that this is still a convex compact subset of $\ell^2$ of infinite dimension. Wikipedia state that every infinite-dimensional convex compact subset of $\ell^2$ is homeomorphic to the Hilbert cube.